The HSL samples were fabricated on (001)-oriented SrTiO\(_3\)  (STO) substrates by pulsed laser deposition with \(in\)  \(situ\)  reflection high energy electron diffraction (RHEED). The details of the growth conditions can be found in Ref.[1]}, [2]}, [3]}, [4]}. The HSL consists of 20 hybrid supercells by monitoring the intensity of RHEED patterns. Single-crystal x-ray diffraction (XRD) measurements were carried out on a Panalytical X'pert MRD diffractometer with wavelength of 1.54 Å at room temperature. Synchrotron XRD measurements and RXMS measurements were performed at the 33-BM-C and 4-ID-D beamlines, respectively, at the Advanced Photon Source of the Argonne National Laboratory. The in-plane resistivity was measured by the standard four-probe method with the Physical Property Measurement System from Quantum Design. The magnetization measurements were performed using the Magnetic Property Measurement System with the Vibrating Sample Magnetometer mode from Quantum Design.
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